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Báo cáo toán học: "Generating functions for permutations which contain a given descent set"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Generating functions for permutations which contain a given descent set. | Generating functions for permutations which contain a given descent set Jeffrey Remmel Department of Mathematics University of California San Diego La Jolla CA 92093-0112. USA j remmel@ucsd.edu Manda Riehl Department of Mathematics University of Wisconsin Eau Claire Eau Claire WI 54702. USA riehlar@uwec.edu Submitted Jan 30 2009 Accepted Feb 7 2010 Published Feb 15 2010 Mathematics Subject Classification 05A15 68R15 06A07 Abstract A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular a large number of generating functions involving the number of descents of a permutation ơ des ơ arise in this way. For any given finite set S of positive integers we develop a method to produce similar generating functions for the set of permutations of the symmetric group Sn whose descent set contains S. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions. Keywords ribbon Schur functions descent sets generating functions permutation statistics 1 Introduction There has been a long line of research 2 3 1 8 9 12 13 14 16 11 which shows that a large number of generating functions for permutation statistics can be obtained by applying homomorphisms defined on the ring of symmetric functions A to simple symmetric function identities. For example the n-th elementary symmetric function en and the n-th homogeneous symmetric function hn are defined by the generating functions E t enE JJ 1 Xit n 0 i and H t h c n r-b x n 0 i i THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R27 1 We let P t Xn 0 pntn where pn Xi xn is the n-th power symmetric function. For any partition p P1 . pf we let X niLi h i eM ni i eR and PM ni i p i . Now it is well known that H t 1 E -t 1.1 and pf. _ En 1 -1 n-1nentn P t E -t 1.2 A surprisingly large number of results on generating functions for various permutation statistics that have appeared in .